The relationship between real-time and discrete-generation models of epidemic spread

Many important results in stochastic epidemic modelling are based on the Reed-Frost model or on other similar models that are characterised by unrealistic temporal dynamics. Nevertheless, they can be extended to many other more realistic models thanks to an argument first provided by Ludwig [Final size distributions for epidemics. Math. Biosci. 23 (1975) 33-46], that states that, for a disease leading to permanent immunity after recovery, under suitable conditions, a continuous-time infectious process has the same final size distribution as another more tractable discrete-generation contact process; in other words, the temporal dynamics of the epidemic can be neglected without affecting the final size distribution. Despite the importance of such an argument, its presence behind many results is often not clearly stated or hidden in references to previous results. In this paper, we reanalyse Ludwig's result, highlighting some of the conditions under which it does not hold and providing a general framework to examine the differences between the continuous-time and the discrete-generation process.